What negative impact has there been on the speed of the process from the current system? Did not all your posted garbage get introduced, like, ah immediately?
Most proposals in my current term have been introduced by friendly Senators, yes, although not all of them (the Anti-Pestering Act was not introduced for a very long time until X suddenly realized it was there) and many bills I've had to specifically ask Senators to introduce as I was writing them. However, during my Vice Presidential term in 2010, many proposals were not introduced, either at all, or were modified by the Senator that introduced them, over the Administration's wishes, and not given a fair debate.
Moreover, removing this procedural requirement eases the burden on my behalf in writing legislation in the first place. If I know I don't have to worry about "shopping" my proposals around in private as or after they are written, I'm more likely to just bang a proposal out in an evening and put it up, knowing that it's immediately in the queue with no fuss and no muss.